Question

A rectangular field is to be fenced on three sides with 1000 m of fencing. The fourth side is a straight river's edge that will not be fenced. Find the dimensions of the field so that the area of the enclosure is 120000 square meters.

Answer 1

length = 500 m , width = 250 m

Explanation:

Let the length of the field is L and the width is W.

Area of the field , A = 120000 sq. metre

Length of fencing = 1000 m

Total length of fence = L + 2 W = 1000

L = 1000 - 2 W

Area = Length x width

A = L x W

A = (1000 - 2W) x W

A = 1000 W - 2 W²

for mxima and minima,

dA/dW = 0

1000 - 4W = 0

W = 250 m

L = 1000 - 2 x 250 = 500 m

Thus, the length of the field is 500 m and the width of the field is 250 m.

Answer 2

`300m* 400m` or
`200m* 600m`

Explanation:

We are given that

Fencing used on three sides=1000 m

Area of field enclosure=120000 square meters

Let x be the length and y be the width of rectangle.

Fencing used on three sides=2x+y

`2x+y=1000`

`y=1000-2x`

Area of field=
`xy=x(1000-2x)`

`-2x^2+1000x=120000`

`-x^2+500x=60000`

`x^2-500x+60000=0`

Using quadratic formula

`x=(-b\pm√(b^2-4ac))/(2a)`

`x=(500\pm√((500)^2-4(60000)))/(2)`

`x=(500\pm 100)/(2)`

`x=(500+100)/(2)=300m`

`x=(500-100)/(2)=200 m`

`y=1000-2(300)=400 m`

`y=1000-2(200)=600m`

Dimension of the field

`300m* 400m` or
`200m* 600m`